Please do the exercise 7 on page 206, the other images may be helpful to you. Thanks for you time, and please write your answer organized and legible.
se 7
cond Recursion theorem The s xed-point theorem rtabe functions for w have an inu of is apearance, theorem in the previous chapter, shows that for any extensional total computable function f there is a number n such that The first Recursion theorem, together with the Myhill-Shepherdson one The second Recursion theorem says that there is such an n even when fis not extensional: we shall prove this in 8 1 of this chapter. aper ried oint or in P.)a This theorem (and its proof) may seem a little strange at first. Never- computability. We shall use it in the promised proof of Myhills theorem e 1 aysisthat theres (theorem 9-3.5) and in the proof of the Speed-up theorem in the next theless it plays an important role in more advanced parts of the theory ofagnmP ef(when computi chapter theorem, we describe some applications and interpretations of it; 82 istoluvenience, an devoted to a discussion of the idea underlying the proof of the theorem, and other matters, including the relationship between the two Recursion In $1, after proving the simplest version of the second Recursionursion theorem is la ro fud oint value for f d tmay e hought the of eurmt he particular numb sion of the proot shows naond the computability theorems. A more general version of the second Recursion theoems her ethoughtth proved in $ 3, and is used to give the proof of Myhill's theorem. 1. stand it. 1.1. Theorem (The second Recursion theorem) such that The second Recursion theorem First let us prove the theorem, and then see how we can ure depends in any essenti dere, heorem 1.1 can b Letf be a total unary computable function; then there is a numbern to ar e underdiying tt e teneral Proof. By the s-m-n theorem there is a total computable function str such that for all x cond Recursion theorem The s xed-point theorem rtabe functions for w have an inu of is apearance, theorem in the previous chapter, shows that for any extensional total computable function f there is a number n such that The first Recursion theorem, together with the Myhill-Shepherdson one The second Recursion theorem says that there is such an n even when fis not extensional: we shall prove this in 8 1 of this chapter. aper ried oint or in P.)a This theorem (and its proof) may seem a little strange at first. Never- computability. We shall use it in the promised proof of Myhills theorem e 1 aysisthat theres (theorem 9-3.5) and in the proof of the Speed-up theorem in the next theless it plays an important role in more advanced parts of the theory ofagnmP ef(when computi chapter theorem, we describe some applications and interpretations of it; 82 istoluvenience, an devoted to a discussion of the idea underlying the proof of the theorem, and other matters, including the relationship between the two Recursion In $1, after proving the simplest version of the second Recursionursion theorem is la ro fud oint value for f d tmay e hought the of eurmt he particular numb sion of the proot shows naond the computability theorems. A more general version of the second Recursion theoems her ethoughtth proved in $ 3, and is used to give the proof of Myhill's theorem. 1. stand it. 1.1. Theorem (The second Recursion theorem) such that The second Recursion theorem First let us prove the theorem, and then see how we can ure depends in any essenti dere, heorem 1.1 can b Letf be a total unary computable function; then there is a numbern to ar e underdiying tt e teneral Proof. By the s-m-n theorem there is a total computable function str such that for all x
